Properties

Label 76.3.b.b
Level $76$
Weight $3$
Character orbit 76.b
Analytic conductor $2.071$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} -\beta_{7} q^{3} + \beta_{5} q^{4} + \beta_{12} q^{5} + ( \beta_{6} - \beta_{12} ) q^{6} + ( -\beta_{2} - \beta_{8} - \beta_{13} ) q^{7} + ( -3 - \beta_{2} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{8} + ( -6 + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} -\beta_{7} q^{3} + \beta_{5} q^{4} + \beta_{12} q^{5} + ( \beta_{6} - \beta_{12} ) q^{6} + ( -\beta_{2} - \beta_{8} - \beta_{13} ) q^{7} + ( -3 - \beta_{2} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{8} + ( -6 + \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{9} + ( -1 - \beta_{7} - \beta_{10} ) q^{10} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{11} + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{13} ) q^{12} + ( 5 - 2 \beta_{5} - \beta_{8} - \beta_{12} + \beta_{13} ) q^{13} + ( 3 + \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{14} + ( -1 + 2 \beta_{1} + 3 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{15} + ( 6 - \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{16} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{10} ) q^{17} + ( 3 - 2 \beta_{1} - 4 \beta_{3} - \beta_{7} + \beta_{10} - 2 \beta_{11} - 2 \beta_{13} ) q^{18} -\beta_{2} q^{19} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{20} + ( -1 + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + 4 \beta_{12} + 2 \beta_{13} ) q^{21} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} ) q^{22} + ( -1 + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{23} + ( -8 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{7} + \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{24} + ( -5 - \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{25} + ( 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} + 3 \beta_{12} + 3 \beta_{13} ) q^{26} + ( 1 + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} + 6 \beta_{7} - \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} ) q^{27} + ( 4 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} + \beta_{11} + 6 \beta_{12} + 5 \beta_{13} ) q^{28} + ( 1 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - 4 \beta_{13} ) q^{29} + ( -16 + 7 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} - \beta_{6} + 5 \beta_{7} + 3 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{13} ) q^{30} + ( 1 - 2 \beta_{2} - 7 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{31} + ( 5 - \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{13} ) q^{32} + ( -3 \beta_{3} + 5 \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} ) q^{33} + ( -6 + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - 4 \beta_{12} - 2 \beta_{13} ) q^{34} + ( -1 + 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{35} + ( 8 + 3 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} - \beta_{4} - 6 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 7 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 3 \beta_{12} ) q^{36} + ( 6 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + 4 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{37} + ( -\beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{38} + ( 3 - 6 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 5 \beta_{7} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{39} + ( -14 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} - 2 \beta_{10} + \beta_{11} - 3 \beta_{12} - 3 \beta_{13} ) q^{40} + ( 18 + 2 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} + 7 \beta_{12} + 3 \beta_{13} ) q^{41} + ( 20 - 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{6} - 8 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 5 \beta_{12} + 6 \beta_{13} ) q^{42} + ( -1 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{13} ) q^{43} + ( -6 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 5 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{44} + ( -12 + 2 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} - 6 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 11 \beta_{12} ) q^{45} + ( 7 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - \beta_{7} - 6 \beta_{8} + 6 \beta_{9} + \beta_{10} + 2 \beta_{12} + 2 \beta_{13} ) q^{46} + ( -1 + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{6} + 7 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{47} + ( 23 - 5 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 8 \beta_{7} - 4 \beta_{11} + 6 \beta_{12} - \beta_{13} ) q^{48} + ( -18 + 6 \beta_{1} - 8 \beta_{2} + 6 \beta_{5} + 8 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 4 \beta_{11} - 4 \beta_{12} ) q^{49} + ( -3 + 2 \beta_{1} - 7 \beta_{3} + \beta_{7} - \beta_{10} + 2 \beta_{11} + 2 \beta_{13} ) q^{50} + ( -6 + 2 \beta_{1} - 14 \beta_{2} + 10 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} ) q^{51} + ( -23 + 3 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + \beta_{4} + 9 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} - \beta_{13} ) q^{52} + ( 5 - 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 7 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} ) q^{53} + ( -14 + 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} + 7 \beta_{12} - 3 \beta_{13} ) q^{54} + ( -1 - 2 \beta_{1} + 6 \beta_{2} + 7 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} ) q^{55} + ( -3 + 11 \beta_{2} - 6 \beta_{3} + 8 \beta_{5} - 4 \beta_{6} + \beta_{7} + 10 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} ) q^{56} + ( 3 - \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} ) q^{57} + ( -12 - 2 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 6 \beta_{10} - \beta_{12} - 10 \beta_{13} ) q^{58} + ( 5 - 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 5 \beta_{11} + \beta_{12} + 3 \beta_{13} ) q^{59} + ( 10 + 2 \beta_{1} - 8 \beta_{2} - 16 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 10 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} - 4 \beta_{12} - 8 \beta_{13} ) q^{60} + ( -6 + 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} + 6 \beta_{6} + 2 \beta_{8} - 6 \beta_{10} - 11 \beta_{12} - 8 \beta_{13} ) q^{61} + ( 32 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{62} + ( 1 - 2 \beta_{1} + 10 \beta_{2} + 21 \beta_{3} + \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + 7 \beta_{7} - \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 8 \beta_{11} + 5 \beta_{12} + 4 \beta_{13} ) q^{63} + ( -4 - 5 \beta_{1} + 10 \beta_{2} + 7 \beta_{3} + 5 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + 7 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} ) q^{64} + ( -32 - 2 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 4 \beta_{11} + 8 \beta_{12} + 2 \beta_{13} ) q^{65} + ( -4 + 2 \beta_{1} - 7 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} - 10 \beta_{12} - 5 \beta_{13} ) q^{66} + ( -2 - 6 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 8 \beta_{11} - 6 \beta_{12} - 4 \beta_{13} ) q^{67} + ( 8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} - 6 \beta_{8} + 2 \beta_{9} + 8 \beta_{10} - 2 \beta_{11} + 6 \beta_{12} + 4 \beta_{13} ) q^{68} + ( 7 + 2 \beta_{1} + 6 \beta_{2} + 4 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 5 \beta_{12} + \beta_{13} ) q^{69} + ( 16 - 5 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} - 5 \beta_{6} + 8 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + \beta_{10} + 5 \beta_{11} + 9 \beta_{12} + \beta_{13} ) q^{70} + ( 2 + 6 \beta_{1} + 4 \beta_{2} - 18 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} ) q^{71} + ( 10 + \beta_{1} + 8 \beta_{2} + 8 \beta_{3} + \beta_{4} - 5 \beta_{5} + 3 \beta_{6} + 13 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + \beta_{11} + 13 \beta_{12} + 11 \beta_{13} ) q^{72} + ( 1 + 4 \beta_{1} - 2 \beta_{3} + 10 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{73} + ( -20 + 2 \beta_{1} - 7 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} - 5 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{74} + ( -1 - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{6} - 4 \beta_{7} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} ) q^{75} + ( 1 - 2 \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{10} + \beta_{12} ) q^{76} + ( 22 - 10 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} + 4 \beta_{4} - 20 \beta_{5} - 6 \beta_{6} + 4 \beta_{8} + 6 \beta_{10} + 15 \beta_{12} + 2 \beta_{13} ) q^{77} + ( 17 + 2 \beta_{1} - 18 \beta_{2} - \beta_{3} - 10 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} - 17 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} + \beta_{10} - 10 \beta_{11} - 18 \beta_{12} ) q^{78} + ( -1 + 2 \beta_{1} + 8 \beta_{2} - 5 \beta_{3} - \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} + 6 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} + 3 \beta_{13} ) q^{79} + ( -24 - 4 \beta_{1} + 7 \beta_{2} - 8 \beta_{3} + 2 \beta_{5} - 5 \beta_{6} + 13 \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} + 4 \beta_{12} - 3 \beta_{13} ) q^{80} + ( 25 - 4 \beta_{1} - 2 \beta_{2} - 24 \beta_{3} - 8 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 8 \beta_{12} - 2 \beta_{13} ) q^{81} + ( -24 - 2 \beta_{1} + 7 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} - 14 \beta_{5} - \beta_{6} - 15 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + \beta_{10} - 3 \beta_{11} + 4 \beta_{12} + 9 \beta_{13} ) q^{82} + ( -2 + 6 \beta_{1} - 6 \beta_{2} + 10 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} ) q^{83} + ( -21 + 3 \beta_{1} - \beta_{2} + 10 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 14 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - 18 \beta_{12} + \beta_{13} ) q^{84} + ( 8 - 2 \beta_{2} - 22 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 6 \beta_{10} - 4 \beta_{11} + 3 \beta_{12} + 4 \beta_{13} ) q^{85} + ( -8 - 11 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + 6 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} - 13 \beta_{12} + \beta_{13} ) q^{86} + ( -1 + 8 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{6} + 3 \beta_{7} - 6 \beta_{8} + 4 \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} ) q^{87} + ( 30 + 7 \beta_{1} - 8 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 5 \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{88} + ( -2 - 2 \beta_{1} - 6 \beta_{2} - 13 \beta_{3} - 8 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - \beta_{7} + \beta_{9} - 4 \beta_{10} - \beta_{11} - 5 \beta_{12} - 5 \beta_{13} ) q^{89} + ( 29 + 4 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} + 19 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 7 \beta_{10} + 2 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} ) q^{90} + ( 7 - 6 \beta_{1} - 8 \beta_{2} - 19 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} + 6 \beta_{7} - 6 \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} - 7 \beta_{13} ) q^{91} + ( 6 + 9 \beta_{1} + 10 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 14 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 3 \beta_{12} + 6 \beta_{13} ) q^{92} + ( -8 - 4 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 12 \beta_{6} + 6 \beta_{7} + 2 \beta_{8} - 6 \beta_{9} + 6 \beta_{11} + 18 \beta_{12} + 4 \beta_{13} ) q^{93} + ( -4 + 5 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 6 \beta_{7} - 8 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + 13 \beta_{12} + \beta_{13} ) q^{94} + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{95} + ( -8 - 3 \beta_{1} - 4 \beta_{2} + 25 \beta_{3} - 5 \beta_{4} - 9 \beta_{5} + 5 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} + \beta_{9} - 6 \beta_{10} - \beta_{11} - 9 \beta_{12} + 4 \beta_{13} ) q^{96} + ( 26 - 10 \beta_{1} + 12 \beta_{2} + 22 \beta_{3} + 8 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} + 2 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} + 8 \beta_{10} + 2 \beta_{11} + 8 \beta_{12} + 2 \beta_{13} ) q^{97} + ( 4 + 8 \beta_{1} - 20 \beta_{2} - 22 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 8 \beta_{7} - 4 \beta_{10} - 4 \beta_{11} - 8 \beta_{12} ) q^{98} + ( 3 + 4 \beta_{1} - 23 \beta_{2} - 11 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 2q^{2} + 2q^{4} - 40q^{8} - 68q^{9} + O(q^{10}) \) \( 14q + 2q^{2} + 2q^{4} - 40q^{8} - 68q^{9} - 12q^{10} + 4q^{12} + 54q^{13} + 30q^{14} + 58q^{16} + 34q^{17} + 36q^{18} + 32q^{20} - 38q^{21} + 36q^{22} - 98q^{24} - 86q^{25} - 16q^{26} + 18q^{28} + 54q^{29} - 204q^{30} + 72q^{32} + 20q^{33} - 82q^{34} + 96q^{36} + 100q^{37} - 148q^{40} + 224q^{41} + 224q^{42} - 96q^{44} - 168q^{45} + 46q^{46} + 296q^{48} - 220q^{49} - 58q^{50} - 288q^{52} + 14q^{53} - 128q^{54} + 12q^{56} + 38q^{57} - 72q^{58} + 188q^{60} + 28q^{61} + 396q^{62} - 118q^{64} - 472q^{65} - 32q^{66} + 30q^{68} + 122q^{69} + 156q^{70} + 80q^{72} + 70q^{73} - 224q^{74} + 228q^{77} + 274q^{78} - 348q^{80} + 334q^{81} - 400q^{82} - 216q^{84} + 48q^{85} - 124q^{86} + 472q^{88} + 416q^{90} + 126q^{92} - 176q^{93} - 88q^{94} - 106q^{96} + 308q^{97} + 68q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{12} - 2 \nu^{11} + \nu^{10} + 14 \nu^{9} - 42 \nu^{8} + 28 \nu^{7} + 132 \nu^{6} - 440 \nu^{5} + 528 \nu^{4} + 448 \nu^{3} - 640 \nu^{2} + 3584 \nu + 1024 \)\()/1024\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{13} + 14 \nu^{12} - 9 \nu^{11} + 94 \nu^{10} - 286 \nu^{9} + 844 \nu^{8} - 980 \nu^{7} - 1880 \nu^{6} + 8592 \nu^{5} - 8832 \nu^{4} - 3968 \nu^{3} + 59392 \nu^{2} - 119808 \nu + 114688 \)\()/20480\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{13} + 2 \nu^{12} - \nu^{11} - 14 \nu^{10} + 42 \nu^{9} - 28 \nu^{8} - 132 \nu^{7} + 440 \nu^{6} - 528 \nu^{5} - 448 \nu^{4} + 2688 \nu^{3} - 3584 \nu^{2} - 1024 \nu + 8192 \)\()/4096\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 1792 \nu^{2} + 4096 \)\()/2048\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} + 1792 \nu^{2} - 8192 \nu + 6144 \)\()/2048\)
\(\beta_{6}\)\(=\)\((\)\( -9 \nu^{13} + 6 \nu^{12} - \nu^{11} + 86 \nu^{10} - 254 \nu^{9} + 476 \nu^{8} - 20 \nu^{7} - 3320 \nu^{6} + 4688 \nu^{5} - 2048 \nu^{4} - 16512 \nu^{3} + 48128 \nu^{2} - 64512 \nu - 22528 \)\()/10240\)
\(\beta_{7}\)\(=\)\((\)\( 21 \nu^{13} + 6 \nu^{12} - 91 \nu^{11} + 246 \nu^{10} - 354 \nu^{9} - 244 \nu^{8} + 2100 \nu^{7} - 1560 \nu^{6} - 112 \nu^{5} + 3392 \nu^{4} - 10112 \nu^{3} + 4608 \nu^{2} - 13312 \nu + 172032 \)\()/20480\)
\(\beta_{8}\)\(=\)\((\)\( 9 \nu^{13} - 166 \nu^{12} - 159 \nu^{11} + 1354 \nu^{10} - 2146 \nu^{9} - 1116 \nu^{8} + 14740 \nu^{7} - 31240 \nu^{6} + 2992 \nu^{5} + 98048 \nu^{4} - 217728 \nu^{3} + 161792 \nu^{2} + 402432 \nu - 1052672 \)\()/20480\)
\(\beta_{9}\)\(=\)\((\)\( 19 \nu^{13} - 126 \nu^{12} - 189 \nu^{11} + 1074 \nu^{10} - 1006 \nu^{9} - 2556 \nu^{8} + 15020 \nu^{7} - 22280 \nu^{6} - 11408 \nu^{5} + 107968 \nu^{4} - 167808 \nu^{3} + 26112 \nu^{2} + 535552 \nu - 991232 \)\()/20480\)
\(\beta_{10}\)\(=\)\((\)\( 39 \nu^{13} + 114 \nu^{12} - 569 \nu^{11} + 674 \nu^{10} + 954 \nu^{9} - 7036 \nu^{8} + 12380 \nu^{7} + 1400 \nu^{6} - 49168 \nu^{5} + 99648 \nu^{4} - 57728 \nu^{3} - 163328 \nu^{2} + 453632 \nu - 69632 \)\()/20480\)
\(\beta_{11}\)\(=\)\((\)\( 11 \nu^{13} - 38 \nu^{12} + 27 \nu^{11} + 106 \nu^{10} - 318 \nu^{9} + 180 \nu^{8} + 1036 \nu^{7} - 3176 \nu^{6} + 2288 \nu^{5} + 4288 \nu^{4} - 11904 \nu^{3} + 2560 \nu^{2} + 46080 \nu - 77824 \)\()/4096\)
\(\beta_{12}\)\(=\)\((\)\( -3 \nu^{13} + 9 \nu^{12} - \nu^{11} - 39 \nu^{10} + 112 \nu^{9} - 82 \nu^{8} - 328 \nu^{7} + 1012 \nu^{6} - 952 \nu^{5} - 1168 \nu^{4} + 4672 \nu^{3} - 4736 \nu^{2} - 7680 \nu + 12288 \)\()/1024\)
\(\beta_{13}\)\(=\)\((\)\( -43 \nu^{13} - 18 \nu^{12} + 293 \nu^{11} - 578 \nu^{10} - 98 \nu^{9} + 3292 \nu^{8} - 8140 \nu^{7} + 3400 \nu^{6} + 22416 \nu^{5} - 56256 \nu^{4} + 50816 \nu^{3} + 78336 \nu^{2} - 308224 \nu + 57344 \)\()/10240\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} + \beta_{4} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_{1} - 5\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta_{1} + 8\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(3 \beta_{13} + 4 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} + 11 \beta_{2} - 7 \beta_{1} + 18\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{13} + 2 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 7 \beta_{8} + 9 \beta_{7} - 17 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + \beta_{1} - 20\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(7 \beta_{13} + 4 \beta_{12} - \beta_{11} - \beta_{10} + 17 \beta_{9} - 11 \beta_{8} + 39 \beta_{7} + 9 \beta_{6} + 15 \beta_{5} + 2 \beta_{4} - 40 \beta_{3} - 25 \beta_{2} + 9 \beta_{1} + 2\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-23 \beta_{13} - 54 \beta_{12} - 53 \beta_{11} - 35 \beta_{10} + 29 \beta_{9} - 33 \beta_{8} - 23 \beta_{7} + 39 \beta_{6} - \beta_{5} - 34 \beta_{4} + 82 \beta_{3} - 53 \beta_{2} + 29 \beta_{1} - 188\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-69 \beta_{13} + 52 \beta_{12} + 19 \beta_{11} - 61 \beta_{10} + 29 \beta_{9} - 15 \beta_{8} - 37 \beta_{7} + 5 \beta_{6} + 159 \beta_{5} - 34 \beta_{4} - 56 \beta_{3} + 59 \beta_{2} - 11 \beta_{1} + 254\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-75 \beta_{13} + 210 \beta_{12} + 127 \beta_{11} - 103 \beta_{10} - 55 \beta_{9} + 115 \beta_{8} + 149 \beta_{7} - 101 \beta_{6} - 133 \beta_{5} + 38 \beta_{4} + 490 \beta_{3} + 255 \beta_{2} - 31 \beta_{1} - 316\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(23 \beta_{13} + 340 \beta_{12} + 223 \beta_{11} - 97 \beta_{10} + 33 \beta_{9} + 5 \beta_{8} + 311 \beta_{7} + 57 \beta_{6} - 189 \beta_{5} - 130 \beta_{4} - 888 \beta_{3} - 89 \beta_{2} + 281 \beta_{1} + 1886\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(137 \beta_{13} - 86 \beta_{12} - 85 \beta_{11} + 61 \beta_{10} + 317 \beta_{9} - 481 \beta_{8} + 201 \beta_{7} + 775 \beta_{6} - 1513 \beta_{5} + 414 \beta_{4} - 1118 \beta_{3} - 373 \beta_{2} + 213 \beta_{1} - 2684\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-781 \beta_{13} - 620 \beta_{12} - 405 \beta_{11} - 453 \beta_{10} - 75 \beta_{9} - 519 \beta_{8} - 1645 \beta_{7} + 189 \beta_{6} + 1567 \beta_{5} - 1290 \beta_{4} - 4648 \beta_{3} - 109 \beta_{2} + 909 \beta_{1} - 6618\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1
−1.92254 + 0.551226i
−1.92254 0.551226i
−1.89728 + 0.632718i
−1.89728 0.632718i
−0.0607713 + 1.99908i
−0.0607713 1.99908i
0.645572 + 1.89294i
0.645572 1.89294i
0.711746 + 1.86907i
0.711746 1.86907i
1.57398 + 1.23393i
1.57398 1.23393i
1.94929 + 0.447510i
1.94929 0.447510i
−1.92254 0.551226i 0.644704i 3.39230 + 2.11950i −2.32715 0.355377 1.23947i 8.62924i −5.35350 5.94475i 8.58436 4.47404 + 1.28279i
39.2 −1.92254 + 0.551226i 0.644704i 3.39230 2.11950i −2.32715 0.355377 + 1.23947i 8.62924i −5.35350 + 5.94475i 8.58436 4.47404 1.28279i
39.3 −1.89728 0.632718i 5.34370i 3.19934 + 2.40089i 5.79268 −3.38106 + 10.1385i 5.87536i −4.55095 6.57943i −19.5551 −10.9903 3.66514i
39.4 −1.89728 + 0.632718i 5.34370i 3.19934 2.40089i 5.79268 −3.38106 10.1385i 5.87536i −4.55095 + 6.57943i −19.5551 −10.9903 + 3.66514i
39.5 −0.0607713 1.99908i 5.37609i −3.99261 + 0.242973i −5.82257 10.7472 0.326712i 5.45132i 0.728358 + 7.96677i −19.9023 0.353845 + 11.6398i
39.6 −0.0607713 + 1.99908i 5.37609i −3.99261 0.242973i −5.82257 10.7472 + 0.326712i 5.45132i 0.728358 7.96677i −19.9023 0.353845 11.6398i
39.7 0.645572 1.89294i 0.820457i −3.16647 2.44406i −2.38184 −1.55308 0.529664i 12.3764i −6.67066 + 4.41614i 8.32685 −1.53765 + 4.50868i
39.8 0.645572 + 1.89294i 0.820457i −3.16647 + 2.44406i −2.38184 −1.55308 + 0.529664i 12.3764i −6.67066 4.41614i 8.32685 −1.53765 4.50868i
39.9 0.711746 1.86907i 4.44946i −2.98683 2.66060i 4.97973 −8.31634 3.16688i 12.2628i −7.09872 + 3.68892i −10.7977 3.54430 9.30745i
39.10 0.711746 + 1.86907i 4.44946i −2.98683 + 2.66060i 4.97973 −8.31634 + 3.16688i 12.2628i −7.09872 3.68892i −10.7977 3.54430 + 9.30745i
39.11 1.57398 1.23393i 2.90118i 0.954817 3.88437i 3.66290 3.57986 + 4.56639i 1.93414i −3.29019 7.29209i 0.583162 5.76533 4.51977i
39.12 1.57398 + 1.23393i 2.90118i 0.954817 + 3.88437i 3.66290 3.57986 4.56639i 1.93414i −3.29019 + 7.29209i 0.583162 5.76533 + 4.51977i
39.13 1.94929 0.447510i 3.19988i 3.59947 1.74465i −3.90374 −1.43198 6.23749i 2.64664i 6.23566 5.01164i −1.23921 −7.60953 + 1.74696i
39.14 1.94929 + 0.447510i 3.19988i 3.59947 + 1.74465i −3.90374 −1.43198 + 6.23749i 2.64664i 6.23566 + 5.01164i −1.23921 −7.60953 1.74696i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.b.b 14
3.b odd 2 1 684.3.g.b 14
4.b odd 2 1 inner 76.3.b.b 14
8.b even 2 1 1216.3.d.d 14
8.d odd 2 1 1216.3.d.d 14
12.b even 2 1 684.3.g.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.b 14 1.a even 1 1 trivial
76.3.b.b 14 4.b odd 2 1 inner
684.3.g.b 14 3.b odd 2 1
684.3.g.b 14 12.b even 2 1
1216.3.d.d 14 8.b even 2 1
1216.3.d.d 14 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 97 T_{3}^{12} + 3595 T_{3}^{10} + 63443 T_{3}^{8} + 539872 T_{3}^{6} + 1940896 T_{3}^{4} + 1665792 T_{3}^{2} + 393984 \) acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16384 - 8192 T + 1024 T^{2} + 3584 T^{3} - 2688 T^{4} + 448 T^{5} + 528 T^{6} - 440 T^{7} + 132 T^{8} + 28 T^{9} - 42 T^{10} + 14 T^{11} + T^{12} - 2 T^{13} + T^{14} \)
$3$ \( 393984 + 1665792 T^{2} + 1940896 T^{4} + 539872 T^{6} + 63443 T^{8} + 3595 T^{10} + 97 T^{12} + T^{14} \)
$5$ \( ( -13312 - 8400 T + 1348 T^{2} + 1337 T^{3} - 28 T^{4} - 66 T^{5} + T^{7} )^{2} \)
$7$ \( 46106276299 + 23021007551 T^{2} + 3578469343 T^{4} + 212956211 T^{6} + 5808593 T^{8} + 75725 T^{10} + 453 T^{12} + T^{14} \)
$11$ \( 94488337600 + 207279189936 T^{2} + 72889781124 T^{4} + 2781706433 T^{6} + 41107428 T^{8} + 280822 T^{10} + 880 T^{12} + T^{14} \)
$13$ \( ( -1265920 + 2184704 T - 907488 T^{2} + 7824 T^{3} + 9959 T^{4} - 293 T^{5} - 27 T^{6} + T^{7} )^{2} \)
$17$ \( ( -69101055 - 18411921 T + 417581 T^{2} + 355811 T^{3} + 9955 T^{4} - 1107 T^{5} - 17 T^{6} + T^{7} )^{2} \)
$19$ \( ( 19 + T^{2} )^{7} \)
$23$ \( 683970816311296 + 864035000025088 T^{2} + 39218066795008 T^{4} + 551803020800 T^{6} + 2798046987 T^{8} + 5204683 T^{10} + 3881 T^{12} + T^{14} \)
$29$ \( ( -112127472 - 91750704 T - 10060400 T^{2} + 839992 T^{3} + 75791 T^{4} - 2901 T^{5} - 27 T^{6} + T^{7} )^{2} \)
$31$ \( 112006799137177600 + 7005784227446784 T^{2} + 136835964338176 T^{4} + 1042205249536 T^{6} + 3570928640 T^{8} + 5704128 T^{10} + 4048 T^{12} + T^{14} \)
$37$ \( ( 914894720 + 490075456 T - 48142944 T^{2} - 85712 T^{3} + 100072 T^{4} - 1508 T^{5} - 50 T^{6} + T^{7} )^{2} \)
$41$ \( ( 18882293760 + 1089553152 T - 442902400 T^{2} - 1190928 T^{3} + 433472 T^{4} - 1776 T^{5} - 112 T^{6} + T^{7} )^{2} \)
$43$ \( \)\(58\!\cdots\!24\)\( + 9894601275600773376 T^{2} + 51208545244013840 T^{4} + 109716719259377 T^{6} + 112918336940 T^{8} + 57363094 T^{10} + 13244 T^{12} + T^{14} \)
$47$ \( 6486650127800008704 + 2593987636485123072 T^{2} + 16602143603532784 T^{4} + 38825334689281 T^{6} + 44446706748 T^{8} + 26838726 T^{10} + 8204 T^{12} + T^{14} \)
$53$ \( ( 18228603200 + 4696986240 T + 390622104 T^{2} + 10269704 T^{3} - 111981 T^{4} - 6645 T^{5} - 7 T^{6} + T^{7} )^{2} \)
$59$ \( \)\(50\!\cdots\!00\)\( + \)\(25\!\cdots\!04\)\( T^{2} + 4608193003005303072 T^{4} + 3925759870481904 T^{6} + 1663620366787 T^{8} + 336602299 T^{10} + 30513 T^{12} + T^{14} \)
$61$ \( ( 522558358600 - 16277769076 T - 1000269562 T^{2} + 29414873 T^{3} + 485080 T^{4} - 14158 T^{5} - 14 T^{6} + T^{7} )^{2} \)
$67$ \( \)\(29\!\cdots\!76\)\( + \)\(25\!\cdots\!00\)\( T^{2} + 5120555266235997504 T^{4} + 3973974955017136 T^{6} + 1503100062371 T^{8} + 293178683 T^{10} + 27889 T^{12} + T^{14} \)
$71$ \( \)\(49\!\cdots\!00\)\( + \)\(57\!\cdots\!76\)\( T^{2} + 15805923995875204096 T^{4} + 11221518796559104 T^{6} + 3467793570368 T^{8} + 526066128 T^{10} + 37812 T^{12} + T^{14} \)
$73$ \( ( -77971926925 - 6681670753 T + 6909415 T^{2} + 11130931 T^{3} + 95497 T^{4} - 5907 T^{5} - 35 T^{6} + T^{7} )^{2} \)
$79$ \( \)\(18\!\cdots\!00\)\( + \)\(22\!\cdots\!64\)\( T^{2} + 7079503618001244160 T^{4} + 6799300948930560 T^{6} + 2745532366528 T^{8} + 498656432 T^{10} + 38740 T^{12} + T^{14} \)
$83$ \( \)\(34\!\cdots\!76\)\( + \)\(85\!\cdots\!08\)\( T^{2} + \)\(17\!\cdots\!88\)\( T^{4} + 81697848347602944 T^{6} + 15885346272448 T^{8} + 1466300592 T^{10} + 62788 T^{12} + T^{14} \)
$89$ \( ( -88310345728 + 32804608000 T - 2083382784 T^{2} + 23700672 T^{3} + 1100800 T^{4} - 23640 T^{5} + T^{7} )^{2} \)
$97$ \( ( 4410892984320 - 293552295936 T - 20455196416 T^{2} + 115812736 T^{3} + 4374632 T^{4} - 27396 T^{5} - 154 T^{6} + T^{7} )^{2} \)
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